int. j. geographical information science, 1997, vol. 11 , no. 8 , 763± 784
Research Article M odelling community evacuation vulnerability using GIS THOMAS J. COVA and RICHARD L. CHURCH National Center for Geographic Information and Analysis (NCGIA), Department of Geography, University of California, Santa Barbara, Santa Barbara, California 93106, USA email:
[email protected] ( Received 26 June 1996; accepted 10 February 1997 ) We present a method for systematically identifying neighbourhoods that may face transportation di Y culties during an evacuation. A classi® cation of this nature o Vers a unique approach to assessing community vulnerability in regions subject to fast-moving hazards of uncertain spatial impact (e.g., urban ® restorms and toxic spills on highways) . The approach is founded on an integer programming ( IP) model called the critical cluster model (CCM ). An heuristic algorithm is described which is capable of producing e Y cient, high-quality solutions to this model in a GIS context. The paper concludes with an application of the method to Santa Barbara, California. Abstract.
1. Introduction
The ® eld of regional evacuation modelling has evolved along a fundamentally temporal line of inquiry, where research has centred on the problem of accurately estimating the time it might take to clear a speci® ed zone of its population. This focus was initially motivated by the perceived threat imposed by nuclear power plants during the 1970s ( WSA 1974 ), and the accidents at Pennsylvania’s Three Mile Island in 1979 and Chernobyl in 1986 served to reinforce this emphasis. The general approach involved prede® ning a circular emergency planning zone ( EPZ) around each nuclear site using a 10-mile radius ( NRC 1980, Urbanik et al . 1980 ) and subsequently pursuing an estimate of the time it might take to clear the zone. Early static-analysis techniques for estimating network clearance time (WSA 1974, Stone 1983) have since been eclipsed by special-purpose transportation simulation models capable of dynamically modelling evacuations on the scale of entire urban areas (SheY et al . 1982, FEMA 1984, Hobeika and Jamei 1985, Pidd et al . 1996). A number of these simulation models have become the basis for evacuation decision support systems ( Han 1990, Tufekci and Kisko 1991, de Silva et al . 1993, Hobeika et al . 1994 ). In the wake of this early evacuation research on nuclear power plants, a modelling paradigm emerged ( Southworth 1991, Urbanik and Jamison 1992). The standard approach was to delimit an EPZ around a known hazard and subsequently apply an evacuation simulation model to explore questions regarding the many factors that might aV ect network clearance time (e.g., routing, population distribution, road and intersection capacity, human behaviour). This general approach proved very useful and has been used to model the evacuation of communities at risk to chemical 1365± 8816/97 $12´00
Ñ
1997 Taylor & Francis Ltd.
764
T . J. Cova and R. L . Church
stockpile sites ( Newsom et al . 1992 ), nuclear research facilities (Sinuany-Stern and Stern 1993 ), dams (Southworth and Chin 1987 ), and hurricanes ( Hobeika et al . 1985 ). A key concept that underlies this modelling approach is the notion of a credible EPZ. A credible EPZ is a valuable spatial construct, as it provides a crisp answer to the dual questions of who needs to be evacuated ( population in the zone) and where they need to be routed to reach safety (outside the zone). In a sense, it serves as a formal agreement among emergency planners regarding the de® nition of a likely evacuation. This allows analysts to move directly to issues related to estimating and reducing the time it might take to clear a zone. As Sorensen et al . ( 1992) note, delimiting a credible EPZ can be a signi® cant political and technical endeavour for certain hazard types. However, a problem arises when an analyst is faced with performing an evacuation assessment for a region that is subject to a hazard with a high degree of uncertainty in its spatial impact. In short, there are many hazards where the population to be evacuated simply cannot be determined in advance. Urban ® restorms, toxic spills on highways, and many other hazards routinely result in ad hoc evacuation zones that are established at the time of the event. For this reason, hazards with a high degree of spatial uncertainty pose an interesting modelling problem. We call this problem the indeterminable EPZ problem ( IEPZ) and state it as follows: How can an evacuation assessment be performed when the population to evacuate is an unknown (i.e., when a credible EPZ cannot be established in advance)? As the IEPZ problem is a spatial problem, it represents a signi® cant opportunity to utilize a GIS approach. The potential role for GIS in evacuation research has been noted by a number of authors (Gatrell and Vincent 1991, Dangermond 1991, Johnson 1992, Rejeski 1993 ), but little work has been done in this area to date. GIS have been applied in generating alternative evacuation routes out of a given zone ( Dangermond 1985, Dunn 1992) and in managing the spatial data associated with an evacuation decision support system (de Silva et al . 1993 ). In general, the wider application of GIS in hazards research has focused on modelling the physical aspects of hazards ( Wadge 1988, Chou 1992, Shu-Quiang and Unwin 1992, Carrara and Guzzetti 1995, Emmi and Horton 1995, Radke 1995) and not on potential evacuation diY culties. Although evacuation vulnerability modelling is clearly related to GIS natural hazards research, it’s more closely aligned with GIS research on modelling human vulnerability and risk (McMaster 1988, Estes et al . 1987, Hodgson and Palm 1992, Burke 1993, Emani et al . 1993, Brainard et al . 1996). The purpose of this paper is to describe a GIS approach to the problem of identifying neighbourhoods that may face transportation diY culties during an evacuation. A secondary concern is to demonstrate that the algorithm underlying this approach is a candidate for addition to the spatial analytic toolbox of contemporary GIS (Goodchild 1987, Burrough 1990, Fotheringham and Rogerson 1993). The paper begins with a description of a methodological framework designed to address the IEPZ problem that we call evacuation vulnerability modelling. A fundamental problem is identi® ed and formulated as an integer programming ( IP) model. Solving this IP model optimally is impractical for most real-world road networks, and we describe a heuristic algorithm designed to produce eY cient, high-quality solutions to the model in a GIS context. An application of the method is presented for Santa Barbara, California, and the paper ® nishes with a conclusion and discussion of further research.
Modelling evacuation vulnerability
765
2. Evacuation vulnerability modelling
An initial approach to addressing the IEPZ problem is to shift the emphasis from a temporal to a spatial perspective. Rather than pursuing the question as to the time it might take to clear a single zone under a particular hazard, regional evacuation can be viewed as a generic process (Perry 1985) independent of any hazard or zone. In other words, clearing people from an area can be viewed as a process independent of any particular zone or hazardous event that might warrant the evacuation. From this perspective, there would be an extremely large number of potential evacuations in any given region. An interesting problem involves classifying them regarding transportation diY culties that might arise during an urgent evacuation (e.g., congestion). Essentially, we require a measure of evacuation diY culty at each point in the plane. One approach is to ® nd a zone for each point that ( 1) contains the point, ( 2) is limited in size, and ( 3) represents the most diY cult evacuation for that point. If this process was performed on a network data model at select points (i.e., intersections), it would be possible to produce a ® eld de® ned along the network that represents an upper bound on potential transportation evacuation diY culties. Goodchild ( 1992 ) would consider this a process of generating a ® eld from a discrete data model. A ® eld of this nature would aid in assessing vulnerability, as communities that are unable to clear their immediate locale in a safe and timely manner may lead to disastrous consequences in some hazard contexts (OFD 1992 ). 2.1. All possible evacuations Regional evacuation is a human process that occurs at the level of the individual, and it is assumed that there are a ® nite number of individuals in any de® ned region at a given point in time. As such, a set theoretic view can be adopted, where an evacuation is de® ned as any subset of a region’s population clearing its immediate locale. The extent of the evacuation is thereby de® ned by the extent of the population involved. With time ® xed (time-slice), there are a ® nite number of potential evacuations, as there are a ® nite number of subsets of any ® nite set. The cardinality of this set is given in equation ( 1) where n is the population of the study area and E is the set of all evacuations. n
|E | =
=
i 1
AB n i
( 1)
For an area with a population as small as 50, there would be greater than 15 10 potential evacuations ( population subsets) at any point in time. A reduction of this set is required to move this problem into a computationally tractable domain (GIS). First, the case where the evacuees are not in proximity is highly unlikely, and contiguity can be added to the de® nition of a valid evacuation. That is, the population must come from a contiguous area. Second, the most critical population to evacuate in the context of fast-moving hazards of uncertain spatial impact is the population within immediate proximity to the origin of the event. To focus on these frequent (Sorensen et al . 1987 ) micro-evacuations, a size limit can be included in the de® nition of a valid evacuation. Lastly, it’s impossible to geo-reference all individuals within a study area, and a common aggregate geographic representation of population must
766
T . J. Cova and R. L . Church
Figure 1.
An example valid evacuation ( black nodes) with two exits ( bold arrows).
be employed. Contiguity, size, and aggregation all serve to signi® cantly reduce the space that comprises all possible evacuations. A common spatial data model for a transportation problem of this nature is a planar network, where the arcs represent road segments and the nodes represent street intersections. One approach to aggregating population in GIS network analysis studies is to assign and aggregate residents to their nearest street intersection using Thiessen polygons centred on the street intersections. An example network data model with one potential evacuation ( black nodes) is given in ® gure 1. We will refer to any arc connecting the set of nodes in the evacuation to the rest of the network as an exit and the set of all exits for a particular evacuation as the exit choice set. An evacuee would be free to choose any arc in the exit choice set as a bridge to `safety’. In graph theory, this set is commonly referred to as a cut-set , as the removal of this edge set results in a separation of the graph into two separate sub-graphs. At this point, the space that comprises all possible evacuations is de® ned as the set of all contiguous node subsets (i.e., intersections or Thiessen centroids) in a transportation network that are less than a given size. 2.2. Evacuation di Y culty Accurately estimating network clearance time for each evacuation using a simulation model is not possible in this context, as there are simply too many cases. However, one viable alternative is to develop a static index estimate of evacuation diY culty. To avoid confusion with estimating network clearance time, we will refer to this index as an evacuation diY culty index. Evacuation diY culty is de® ned as an estimate of the relative eV ort required to clear an area of its population. This concept embodies a composite of the potential for congestion, accidents, and general diY culty in deploying response vehicles into the evacuation zone. There are numerous approaches to operationalizing a concept like evacuation diY culty, but for our purposes, we will simply de® ne it in equation ( 2) where P is the population involved and C is a measure of the capacity of the exit choice set. P/ C
( 2)
It is clear that this de® nition is a highly simpli® ed view of what is undoubtedly a complex social, spatio-temporal process. It should be noted that this method is a comparison of starting scenarios and not predicted outcomes. In other words, the comparison is between initial conditions, rather than an attempt to predict the outcome of an actual evacuation.
Modelling evacuation vulnerability
767
2.3. Spatial evacuation vulnerability With the space that comprises all possible evacuations de® ned and an approach to scoring their potential diY culty, the focus can shift to developing a systematic spatial classi® cation method. One approach is to inquire as to the worst-case (i.e., maximum diY culty) that each node in a network might be involved in, less than a certain size. We call this worst-case for a particular node (and size limit) its spatial evacuation vulnerability . The primary task is, then, to map the local variation in evacuation vulnerability throughout a network due to the unique geographical setting of individual nodes. The de® nition of `worst-case’ in this context refers to the maximum value for a given diY culty measure and not the actual worst-case that might occur. For example, an actual evacuation could easily be more diY cult due to the loss of exits (e.g., hazard or stalled vehicle), signi® cant imbalances in the number of evacuees utilizing various exits, or convergence on the hazard site by response personnel and local citizens. The de® nition of worst-case utilized in this paper is simply an attempt to systematically reveal the geographical context in which an evacuation might take place. An essential element in the de® nition of spatial evacuation vulnerability is the method for limiting evacuation size. In exploring the local neighbourhood of each node, it’s necessary to formalize the notion of `local’. We have identi® ed ® ve approaches to achieving this end: Euclidean distance, network distance, population, area, and node count (® gure 2). In this paper, node count is utilized as the size limit, which assumes that network connectivity relations are the most critical component in de® ning evacuation vulnerability. This approach is particularly useful in areas subject to urban wild® res where there are very long dead-end roads extending into the urban/ wildland interface. If Euclidean distance is utilized to limit evacuation size in this context, these dead-end roads appear as viable exits until the distance had been increased beyond a neighbourhood scale. It is anticipated that these various approaches to limiting evacuation size will have application in diV erent hazard and network contexts, but we leave a thorough investigation of this issue for further research. Figure 3 provides a theoretical example of how spatial evacuation vulnerability would be de® ned for a node, given that node count is the approach to limiting evacuation size. For this example, the evacuation node set must be less than or equal to six nodes. The ® gure shows three potential evacuations that all contain the node labelled A (note: There are many more than three node sets involving node A ). Assume that each node has 100 residents, and each arc has one lane in each direction. The nodes involved are shown in black and the exits are shown as bold arrows. The
Figure 2.
Five approaches to limiting evacuation size.
768
Figure 3.
T . J. Cova and R. L . Church
Three example evacuations (shaded ) less than or equal to six nodes that all involve the population assigned to node A .
area assigned to each node (square Thiessen polygons) is shown with dashed lines, and the area to be cleared is shaded. In case 1, there are 400 evacuees and 2 exits ( 200 evacuees per exit lane). In case 2, there are 600 evacuees and three exits (200 evacuees per exit lane), and in case 3 there are 600 evacuees and only two exits ( 300 evacuees per exit lane). For this reason, case 3 represents node A’s worst-case (maximum population per exit lane) out of the three cases (and in general as well ), and node A’s spatial evacuation vulnerability value would be set to 300 people per lane. The classi® cation of a region based on spatial evacuation vulnerability can then be achieved by iteratively posing the same query to each node in the network. An outline of this process is now o Vered. For a network with population assigned to its nodes and lanes on its arcs: 1. Select a method for limiting evacuation size and set the required parameters. 2. For each node, satisfy the following spatial query: `What is the most diY cult (worst-case) evacuation starting scenario within which residents at this node might participate, where the evacuation is smaller than the speci® ed size limit?’ 3. Assign each arc in the network the higher spatial evacuation vulnerability value of its two end nodes. 4. Classify the arcs according to their spatial evacuation vulnerability value. 3. The critical cluster model
3.1. Backgrou nd The model formulated in this section represents the basis of our approach to modelling evacuation vulnerability. We will refer to the problem as the critical cluster problem (CCP) and the model formulated to solve it as the critical cluster model (CCM ). Each individual node in a street network represents a separate optimization problem under this model. The process is akin to asking each node in the network the question posed in step 2 above. The answer to this question varies spatially as nodes do not have equivalent surroundings. The critical cluster problem involves ® nding potentially the most diY cult evacuation starting scenario associated with a pre-speci® ed node, which we will call the root node. For a given root node and speci® ed cluster size in nodes, there is at least one cluster of nodes in a network that contains the root node and maximizes the
Modelling evacuation vulnerability
Figure 4.
769
A network and a corresponding plot of evacuation di Y culty for the black node.
ratio of cluster population to exit lanes. Solving the critical cluster problem involves ® nding the most extreme of these clusters for a given root node across a range of cluster sizes from one node up to a speci® ed size limit. This optimal cluster (i.e., the cluster containing the root node that has the highest average population per exit lane) is referred to as the critical cluster. An example of a critical cluster is given in ® gure 4 for the black node. To simplify this example, each node has a population of 1 and each arc has one lane in each direction. Note that for the example root node, the critical cluster occurs at a cluster size of ® ve; the point where the diY culty measure reaches its maximum. The problem of ® nding a node’s critical cluster falls within the broad category of graph or network partitioning problems. A partition is a subset of nodes within a larger network, and an optimal partitioning maximizes or minimizes some speci® ed criteria related to the partition. In this context, we are only interested in contiguous partitions which we refer to as clusters. A related problem that is addressed widely in the operations research literature is the graph partitioning problem ( Kernighan and Lin 1970, Johnson et al . 1989, Jin and Chan 1992, Laguna et al . 1994, Pirkul and Rolland 1994 ). This problem was originally stated by Kernighan and Lin ( 1970, p. 291) as: . . . given a graph G with costs on its edges, partition the nodes of G into subsets of no larger that a given maximum size, so as to minimize the total cost of the edges cut.
The graph partitioning problem arises when the goal is to divide a graph into separate sub-graphs with a minimum number of connections between the resultant sub-graphs. This problem is related to the critical cluster problem and represents a valuable starting point from which to derive a formulation. The critical cluster problem can be stated in similar terms as: Given a graph G with costs on its edges and weights on its nodes, ® nd a contiguous partition less than a given maximum size that contains a pre-speci® ed node i *, so as to maximize the total weight of the partition relative to the total cost of the edges cut.
Within the context of the diY culty measure presented in § 2, node weight is population, and edge cost is the number of lanes. Note that an edge may not have the same number of lanes in each direction.
770
T . J. Cova and R. L . Church
3.2. Formulation To specify the critical cluster unambiguously, given a graph G = (N , A ) with costs, c ij ( lanes), on its arcs and weights, a i ( population), on its nodes, we are asked to partition N into two node sets N1 and N 2 such that N = N 1 n N2 , N 1 m N 2 = B, where N 1 is a contiguous partition less than or equal to size s (in nodes) that contains a root node i *, so as to maximize the total weight of N 1 relative to the total cost of the arcs cut between N1 and N 2 . This can be formulated as an integer programming ( IP) problem: Objective :
Maximize:
Subject to :
xiÕ
x j < y ij
i
i j
a ix i
( 3)
c ij y ij
Y i , j and Y j , i × N
xi < s
( 5)
x i*= 1
( 6)
i
x i , y ij , × {0,1} W here :
( 4)
xi=
G G
y ij =
Y i, j × N
( 7)
1, if node i is in N 1 0, otherwise 1, if node i is in N1 and node j is in N 2 0, otherwise
a i= weight of node i ( population) c ij= cost of arc ij ( lanes) s=maximum size (in nodes) of N 1 i *=index of root node × N 1
In this formulation for the CCM, the objective function (3 ) maximizes the ratio of the partition’s total weight to the total cost of the arcs with one node in the partition and one node outside the partition. Constraint ( 4 ) ensures that if node i is in the partition and node j is not, then arc y ij must be equal to 1, as it is a connection between the partition containing node i * and the rest of the graph. If an arc’s y ij value is equal to 1, then the arc’s number of lanes is included in the denominator of the objective function and the total number of exit lanes for the partition. Constraint ( 5) limits the search to partitions less than a speci® ed size in nodes. Constraint ( 6 ) states that node i * must be in the partition node set N1 . Constraint ( 7) ensures that all y ij and x i variables are binary integer variables. There is also the condition that the critical cluster be contiguous. If we solve the above problem and the optimal critical cluster is, in fact, a multiple cluster, then additional constraints are required to ensure contiguity. Note that this problem is also a nonlinear problem, as the objective involves optimizing a ratio of two terms each with variable terms. Although we have developed an involved process for deriving optimal solutions to the CCM, ® nding guaranteed optimal solutions is impractical for most real world street networks. Because the goal is to solve the CCM in a practical GIS application context (i.e., arbitrarily large street networks in user time), the next section describes a heuristic algorithm designed to produce good solutions to this model in an eY cient manner.
Modelling evacuation vulnerability
771
4. Heuristic algorithm
4.1. Design Conceptually, the heuristic `grows’ a cluster from a speci® ed root node in a network in an attempt to locate the node’s critical cluster. Beginning with the speci® ed node, the heuristic iteratively adds one of the currently adjacent nodes to the existing contiguous cluster. In constructing a cluster of nodes from a root node, there is one signi® cant decision to make at each iteration: which adjacent node should be added next? A node must be assessed by the population it will add to the cluster, as well as its net eV ect on the cluster’s total exit capacity. In the context of this paper, we have de® ned an arc’s capacity simply to be its number of lanes. The equation for a node’s gain regarding the objective value is given in equation ( 8):
+1= Pk + a i = g i Pk or g i = Ck (Pk Õ a i ) Ck Pk (Ck + (o i Õ c i )) +1 C k + (o i Õ c i )
Pk Ck
( 8)
W here: k =index of iteration
g i= gain in the objective if node i is selected Pk= total population of cluster at iteration k C k= total exit capacity of cluster at iteration k a i= population at node i o i= new exit capacity node i would open, if selected c i= existing exit capacity node i would close, if selected
Figure 5 shows an example of growing a cluster from a root node in an attempt to ® nd the critical cluster. To simplify this example, each node has a population of 1, and each arc has 1 lane in each direction. At step 1, the cluster began as one node (root node shown in black) with three exit lanes, giving it an objective value of 1/ 3. As there are three lanes leading out of the cluster, there are three adjacent nodes to evaluate with the above calculated gain value. At step 2, the node that most increased the objective value was the one to the left, and it is selected to give the cluster an objective value of 2/ 2= 1. This added node was as isolated node (i.e., dead-end street) except for its connection to the root node. The process continues, where any adjacent node to the entire cluster is a candidate node to be added in the next iteration. As this example relied on selecting the adjacent node that most increased the objective value, it is a greedy approach to ® nding the critical cluster. A greedy approach can rapidly lead to local optima, and it is likely that the best cluster found
Figure 5. Growing a cluster from a root node ( black node) in an attempt to ® nd that node’s critical cluster, where each node has unit population and each arc has one lane in each direction.
772
T . J. Cova and R. L . Church
by this approach will not be optimal. However, solution quality can be greatly improved by modifying the procedure to use a semi-greedy approach. In a semigreedy approach, an alpha parameter (expressed as a percentage) is used to increase the possible list of adjacent nodes that might be selected to add to the cluster. In short, the improvement factor for all adjacent nodes is calculated using equation ( 8), and all adjacent nodes are scanned again to produce a list of only the ones that are greater than alpha per cent of the best option. This implies that nodes with gain values less than the best node may be possible candidates for selection. A random selection is made from this list of candidate additions at each step. For this reason, successive runs of the heuristic for a given root node will likely result in diV erent results. A second parameter (starts) can be added to control the number of times each node is run, where the best overall objective value is saved. In addition to semi-greedy selection, we add the concept of adaptiveness found in the GRASP approach ( Laguna et al . 1994). Adaptiveness is an answer to the notion that selecting one adjacent node can change the gain value of other adjacent nodes. In short, selecting one node aV ects the relative scores of all other adjacent nodes. This has a signi® cant impact on the complexity of the algorithm, as it implies that the gain values for all adjacent nodes must be recalculated after every node selection. In this way, the nodes are thought to adapt their respective gain values to the changing state of the cluster. 4.2. Implementation 4.2.1. GIS consideration s The GIS query primitive that this algorithm requires is the forward star ( Evans and Minieka 1992 ). A node’s forward star is the set of arcs directed away from the node, which facilitates the retrieval of the set of immediately adjacent nodes. Figure 6 shows two examples of a forward star, one with undirected arcs (Case A) and one with directed arcs (Case B). The forward star query is utilized in many GIS network analysis algorithms like shortest path, routing, and tracing. As an example, ESRI’s ARC / INFO 7.0 supports the forward star query through the Arc Macro Language (AML) command SHOW with the ADJACENT and ADJACENT NEXT options. The ADJACENT option returns the number of nodes that can be reached via the undirected forward star, where subsequent calls to SHOW with the ADJACENT NEXT option return the node ids. Additional AML code can be added to remove nodes when considering street directionality (arc attribute). Although our ® rst attempt to implement this heuristic relied on this approach, AML is an interpreted macro language and does not allow direct access to the network data structure. As such, it proved too slow for our computational
Figure 6.
Two cases of a forward star ( bold arcs) for the centre node.
Modelling evacuation vulnerability
773
experimental design, as we wanted to be able to grow many clusters from each node in a network of more than 5000 nodes using a variety of alternatives regarding the decision of which node to add at each step. For this reason, we implemented our own forward star network data structure ( Evans and Minieka 1992) in a stand-alone C program and loosely coupled the program with ARC / INFO by exporting the network to text and importing the heuristic results back into ARC / INFO. In an environment where a software engineer has access to the internals of the GIS network data structure, the forward star query could be optimized, and the heuristic could be run in user-time within the GIS. A compromise between these two engineering extremes is ESRI’s ARCVIEW 3.0, which has support for the forward star query through a dynamic link library ( DLL ) that can be called directly from a compiled C program ( Honeycutt 1996 ). 4.2.2. Speci® cation The implementation of the heuristic takes as input a textual representation of a network and a set of parameters, whereby it produces an output ® le of node id’s and their associated spatial evacuation vulnerability values. The parameters of the program are given in table 1. There are two conditions that must be met before a network can be considered valid for an application of the heuristic: contiguity and global exit. Contiguity implies that there must exist at least one path between every pair of nodes in the network, and global exit implies that at least one node in the network be designated as an exit from the entire network. A global exit may not be selected as a component node of any cluster, and as most digital road networks are subsets of a larger network, it should be clear which nodes represent exits from the network. The main control logic behind the algorithm is given in ® gure 7 as pseudocode.
Figure 7.
The pseudocode for the heuristic algorithm.
774
T . J. Cova and R. L . Church
Given a size limit, the algorithm sequentially runs through the nodes in a network, in an attempt to ® nd the critical cluster for each node. The only required parameter is the size limit, as the subsequent parameters are only used to improve the quality of the solutions produced by the algorithm. The total cluster weight over the number of exit lanes is referred to as the weight / cost value. An important bene® t of the structure of a critical cluster, in general, is that all the nodes that comprise one must have a critical cluster weight / cost ratio at least as large as the original critical cluster’s. This means that when a maximum cluster is found for a particular root node, the nodes that comprise this cluster that have a critical cluster weight / cost value less than this critical cluster’s value can automatically be raised up to this new value. It is still possible, however, for a subset of these component nodes to have an even higher critical cluster value. This improvement strategy is listed as step ( 10). 4.3. Evaluation To test the performance of the heuristic algorithm, optimal solutions to the CCM were derived for 40 randomly selected nodes from 4 real-world street networks ( 10 each) at three evacuation size limits 10, 25, and 50 ( 120 problems), where the networks ranged in size from 200 to 300 nodes. The equation for evaluating the per cent from optimal for a node and size limit is: OÕ
B
( 9)
O
where O is the optimal solution and B is the best solution achieved by the heuristic. Table 2 shows the mean per cent from optimal for the 120 problems for each combination of the semi-greedy parameters alpha and starts . A high alpha (e.g., near 1 ) constrains the growth of the cluster to greedy, where lowering alpha results in progressively `wilder’ cluster growth. Note that for a starts of 1 (row 1), the solution Table 1. Name
Range
Size limit Alpha Starts
1 . . . nÕ 0 ... 1 1 ... x
Table 2. Starts 1 2 4 8 16 32 64 128
Program parameters. Description
1
The size limit (in nodes) to terminate growth The semi-greedy percentage parameter The number of times to start the heuristic from a root node
Mean percentage from optimal varying alpha and starts ( 120 problems per cell ). 0´975
12´07 10´57 9´53 7´72 7´72 7´81 6´58 6´58
0´950 10´53 8´56 7´13 6´12 6´12 5´25 4´85 4´74
0´925 11´21 9´32 8´16 5´92 5´92 4´95 4´48 4´13
0´900 11´61 9´14 7´75 5´08 5´08 4´29 3´77 3´95
alpha 0´875
11´01 10´23 7´47 6´73 6´73 4´34 4´32 3´69
0´850 12´69 9´52 8´08 5´92 5´92 4´53 3´84 3´87
0´825 12´04 10´27 8´03 7´35 7´35 4´45 4´15 3´61
0´800 15´14 11´18 8´51 6´82 6´82 5´06 4´2 4´02
0´775 15´42 13´06 9´97 6´65 6´65 5´02 3´81 2´99
0´750 17´67 11´93 8´59 8´04 8´04 5´02 3´93 3´72
Modelling evacuation vulnerability
775
quality decreases as alpha is decreased and the cluster growth gets wilder. Also, for all columns, solution quality improves as starts is increased. The best overall combination occurred at an alpha of 0´775 and a starts of 128, where the average percentage from optimal was 2´99. For our sample set of nodes, this implies that the best combination for high solution quality is moderately wild cluster growth and a large number of starts. In evaluating the solution quality of the heuristic, it is important to keep in mind the intended purpose for the results. The goal is to produce an evacuation vulnerability map with a particular classi® cation scheme within a user time frame. There are two additional concepts that aV ect the required heuristic solution quality necessary to produce an evacuation vulnerability map. Figure 8 shows how a relatively poor performance ( 17´5 per cent from optimal ) of the heuristic for a particular node may not result in a misclassi® cation in the output map for that node. This implies that the number of desired output map classes in¯ uences the required heuristic solution quality to produce a high quality map. The second concept that obviously in¯ uences the required heuristic solution quality is the desired heuristic classi® cation accuracy of the output map. In other words, an output map within 99 per cent of the nodes classi® ed correctly will require more computational eV ort than a map within 90 per cent classi® ed correctly. Putting these two concepts together introduces two additional parameters to the production of an evacuation vulnerability map: desired class granularity and heuristic classi® cation accuracy. As solution quality can be traded for time, we want to select the avenue that results in solution quality suY cient to produce the desired map classi® cation accuracy in the least amount of time. This implies that a two class map (e.g., high vulnerability and low vulnerability) with 90 per cent of the nodes classi® ed correctly will undoubtedly take a fraction of the time it would take to produce a 5 class map with 99 per cent of the nodes classi® ed correctly. To gain insight into the relationship between solution quality and classi® cation accuracy, a simple experiment was devised. A parameter was added that de® nes the number of classes of the ® nal output map, where the classes are de® ned as ® xedintervals between 0 and the highest optimal value across all nodes for a particular evacuation size limit. This facilitates a comparison between the class of the optimal solution for a given node and the class assigned by the heuristic. A node is considered either correctly or incorrectly classi® ed by the heuristic. The overall classi® cation accuracy can be expressed as a per cent de® ned as the total number of correctly classi® ed nodes divided by the total number of nodes ( 120 ). Table 3 shows the values of alpha and starts necessary to produce a map at speci® ed levels of classi® cation accuracy while varying the number of classes. Producing a 5 class map with 99 per cent of the nodes classi® ed correctly was deemed impractical from a computational
Figure 8.
Poor heuristic results for a node may not result in a misclassi® cation.
776
T . J. Cova and R. L . Church
Table 3.
Required (alpha , starts) pairs to produce a map at a speci® ed heuristic classi® cation accuracy for a speci® ed number of classes. Node classi® cation accuracy (per cent )
No. of classes 2 3 4 5
90 ( 0´95, ( 0´95, ( 0´95, ( 0´90,
Figure 9.
1) 1) 8) 32 )
95
97´5
99
(0´95, 1) (0´95, 1) (0´95, 16) (0´775, 64 )
( 0´95, 1) ( 0´95, 8) ( 0´775, 64) ( 0´75, 512)
( 0´95, 8) ( 0´95, 16) ( 0´775, 256) impractical
The relation between arc and node classi® cation accuracy.
perspective, as producing a map with 97´5 of the nodes classi® ed correctly required 512 starts of the heuristic at each node in the network. One ® nal point worth taking up before proceeding to the presentation of the method in an example case-study is arc classi® cation accuracy. As described in § 2, after the nodes of a particular network have been classi® ed, the vulnerability level of an arc is de® ned to be the worst-case vulnerability level of its two end nodes. This implies that the classi® cation accuracy of an arc depends on the classi® cation accuracy of its two end nodes. Figure 9 shows the eight feasible cases that may occur. On the left is a table for the cases where the vulnerability level of node 1 is greater than that of node 2 and the converse. For example, the upper left corner shows that if node 1’s vulnerability level is greater than node 2’s and node 1 is correctly classi® ed then, `Yes, the arc will be correctly classi® ed’. A more interesting table on the right side of ® gure 9 shows that when the vulnerability levels of node 1 and node 2 are equal (a highly likely case), then there is only one case where the arc will not be correctly classi® ed. In short, the classi® cation accuracy of the arcs in a evacuation vulnerability map will always be higher than the classi® cation accuracy of its nodes. As the ® nal map is a plot of the arc vulnerability levels, this is good. Unfortunately, it is impractical to empirically assess the arc classi® cation accuracy of the heuristic because this requires that optimal solutions be known for a very large number of nodes in a network. However, knowledge of this factor allows one to stipulate that the classi® cation accuracy of a particular map exceeds the value of the map’s node classi® cation accuracy (e.g., heuristic classi® cation accuracy greater than 95 per cent). 5. Ex ample case study: Santa Barbara, California
In the last few decades, the residents of Santa Barbara, California have endured toxic spills, ® restorms, ¯ oods, and other hazards that have resulted in numerous
Modelling evacuation vulnerability
777
evacuations. In addition to these fast-moving hazards, whose resulting evacuations could not have been delimited in advance, Santa Barbara is an ideal region to examine regarding evacuation vulnerability due to its wide variety of street patterns and residential con® gurations. This section uses Santa Barbara as a sample region to perform an example case study to highlight some of the issues that arise in a practical application of the method described in the prior sections. 5.1. Preparation An initial hurdle in performing this study involved acquiring the necessary data. There are essentially two classes of geographic information required to perform a case-study, given the diY culty measure presented in § 2: streets and population. Data regarding population for the Santa Barbara area was acquired from 1990 Census Tiger ® les, and data on the area’s roads was provided by Navigation Technologies ( NavTech). Census Tiger street data would also suY ce, but we found NavTech’s data to be of a higher quality. The data preparation for a study is minimal, as the model requires that all nodes have a representative population and all street segments have the number of lanes in each direction. The most involved step was transferring the population from the census data blocks to the nodes of the Navigation Technologies data. One approach to this process is depicted in ® gure 10, where Thiessen polygons are generated around the nodes of the street coverage and overlayed with the census Tiger blocks. The polygon `fragments’ are then each assigned to a node based on their population. A fragment’s population is the fraction of the area within the block multiplied by the block’s population. The shaded pieces in ® gure 10 would be assigned to the node in the upper left. Flowerdew and Green ( 1992 ) have researched this areal interpolation problem (zone-to-zone), and the reader is referred to this work for a complete review. 5.2. Analysis Aside from selecting a measure of evacuation diY culty, the main parameter to the procedure is the size limit (s). Assuming that the measure of diY culty is the one described in § 2, and node count is the selected means for limiting size, a researcher must decide what node count limit to place on the process. In the cases of urban wild® res and toxic spills, the most critical area to clear is the immediate vicinity of the hazard. For the purposes of this example study, this was taken to be a size limit of 25 nodes. As node count is the size limit, `vicinity’ in this case refers to network
Figure 10.
Thiessen polygon population interpolation process.
778
T . J. Cova and R. L . Church
Figure 11.
Figure 12.
An evacuation vulnerability map for the Santa Barbara vicinity.
An evacuation vulnerability map of western Santa Barbara.
vicinity and not Euclidean distance. In other words, two nodes that are relatively far apart in Euclidean distance are considered `close’ if they are connected by a road because the potential for interaction between two evacuees starting from each of these nodes is relatively high.
779
Modelling evacuation vulnerability
Figure 11 shows the results of producing a complete map for the Santa Barbara vicinity. The thematic map unit is the number of people per lane in an road segment’s worst-case evacuation (i.e., maximum diY culty value less than 25 nodes). Conceptually, the map is a discrete surface de® ned only along the network. Because nearby road segments are often in the same spatial evacuation vulnerability class, groups of segments organize themselves into perceivable vulnerability clusters. We call these clusters evacuation sheds , and they represent interesting areas for further inquiry. However, at this mapping scale the network is too dense to reveal why certain neighbourhoods are highlighted. Figure 12 shows a larger scale view of an area just west of the Santa Barbara City limits with many evacuation `hot spots’. Communities (A) through ( D) all have more than 500 residents and only one exit lane. For this reason, there is greater potential for traY c congestion to impede an urgent evacuation of these neighbourhoods. Despite the relatively large number of exits in the larger community ( E), the population density in this area is so high that the algorithm had no problem ® nding numerous potential evacuations where the number of residents per exit lane might be greater than 500. Figure 13 shows a larger scale view of downtown Santa Barbara. A clear evacuation vulnerability corridor is visible along the primary artery (i.e., six lane freeway shown in gray) through this area. This is partly due to the numerous dead-end streets created by the construction of the freeway in addition to the relatively high residential density along this route. A prominent evacuation shed (A) on a relatively sharp corner of the freeway appears in red. As this neighbourhood is adjacent to the freeway, it has almost no connecting streets along its north-east side. Also, on its south-west side of this neighbourhood is a steep hillside (not shown) that results in even fewer connecting streets. Finally, its residential density is higher than other neighbourhoods in this area. 5.3. Discussion As census data was used to assign population to the nodes on the network, these maps can be considered a timeslice of Santa Barbara’s population at night. As in most urban areas, the population distribution during the day, night, and peak commuting hours varies considerably. In addition to population movement, these maps were constructed at one evacuation size limit ( 25 nodes). Table 4 shows a more comprehensive approach to a case-study, where three time periods (daytime, rushhour, night-time) are combined with three size limits ( 25, 50, 100) to produce nine maps for exploring spatio-temporal evacuation vulnerability in this area. Parrot and Stutz ( 1991) have noted the importance of modelling daytime shifts in population in the context of emergency planning, and they provide one example approach for Table 4.
A matrix to produce nine evacuation vulnerability maps for a study area. Time of day ( population)
Evacuation size limit
day-time
night-time
rush hour
25 50 100
1 4 7
2 5 8
3 6 9
780
T . J. Cova and R. L . Church
Figure 13.
An evacuation vulnerability map of downtown Santa Barbara.
San Diego County. Glickman ( 1986 ) and Goodchild et al . ( 1993) have also addressed this problem from diV ering perspectives, but as Southworth ( 1991 ) notes, this is an area in need of further research. 6. Conclusion
The primary objective of this research was to propose one approach to the problem of performing regional evacuation analyses in areas subject to fast-moving hazards of uncertain spatial impact. When a region is subject to a hazard of this nature, it is impossible to delimit a credible emergency planning zone ( EPZ) to apply any one of a host of contemporary evacuation simulation models. To overcome this problem, a new method was presented that allows a researcher to focus on the spatial variation in evacuation diY culty across a landscape. This facilitates a systematic geographical approach to studying community evacuation vulnerability. Evacuation vulnerability modelling represents a proactive perspective for emergency planners and other parties that may be interested in disaster mitigation and preparedness. These maps can be viewed as a new vulnerability layer for hazards research. Risk can be viewed as a function of hazard and vulnerability (Alexander 1993), so if a ® re hazard map represents the hazard component of risk, then an evacuation vulnerability map could be overlayed with this map to produce a risk
Modelling evacuation vulnerability
Figure 14.
781
The potential role for evacuation vulnerability maps in risk mapping.
map. This opens the door to integrating evacuation vulnerability modelling with the numerous GIS hazard models that have been developed. Figure 14 shows example hazard layers and how they might be combined with an evacuation vulnerability layer to explore issues of risk. Another interesting area of research that needs to be addressed in this context is developing new approaches to estimating the whereabouts of population in a city at a relatively ® ne-grained level (e.g., census block) for ® xed points in time. This problem is extremely complex, as population ¯ uctuations range from low-frequency seasonal migrations to the `noise’ of special events. Also, there are also a host of interesting research questions and problems to address in developing new measures of evacuation diY culty and spatial evacuation vulnerability. First, this paper presented only one measure of evacuation diY culty, but there is a clear need to develop additional measures that take into consideration other factors that aV ect evacuation diY culty (e.g., number of vehicles, special populations) ( Vogt and Sorensen 1992). These measures might be considered in the larger family of accessibility measures where they refer to the accessibility of a population subset out of a neighbourhood. Second, the modelling decision of how to limit evacuation size is an interesting area for further research. This paper presented ® ve approaches that may each have application in diV erent hazard or network contexts. An investigation into the strengths and weaknesses of these various approaches for diV erent hazard and network contexts would be a valuable study. Acknowledgments
The NCGIA is supported by the NSF through grant SBR 88± 10917. The authors would like to thank Michael Goodchild for his general support and the following people for their contribution to this project: Uwe Deichmann, Jonathan Gottsegen, Terry Figel, and the IJGIS reviewers. We also wish to thank Navigation Technologies ( NavTech) for their generosity in allowing us access to their database for Santa Barbara County. An earlier version of this paper received the IGIF student paper award at GIS / LIS `95, for which the lead author is especially grateful. The latter part of this research was supported by an Eisenhower Graduate Fellowship from the National Highway Institute ( NHI).
782
T . J. Cova and R. L . Church
References A lexander, D ., 1993, Natural Disasters (New York: Chapman and Hall ). B rainard J ., L ovett A ., and P arfitt J ., 1996, Assessing hazardous waste transport risks using a GIS. International Journal of Geographical Information Systems, 10, 831± 849. B urke, L . M ., 1993, Race and environmental equity: a geographical analysis of Los Angeles. Geo Info Systems, 3, 44± 50. B urrogh, P . A ., 1990, Methods of spatial analysis in GIS. International Journal of Geographical Information Systems, 4, 221± 223. C arrara, A ., and G uzzetti, F . (editors), 1995 , Geographical Information Systems in Assessing Natural Hazards ( Dordrecht: Kluwer Academic Publishers). C hou, Y . H ., 1992, Management of wild® res with a geographical information system. International Journal of Geographical Information Systems, 6, 123± 140. D angermond, J ., 1985, Network allocation modelling for emergency planning. Proceedings of the Conference on Emergency Planning: Emergency Planning, Simulation Series,
Volume 15, Number 1, edited by J. M. Carroll ( La Jolla: Society for Computer Simulation), pp. 101± 106. D angermond, J ., 1991, Applications of GIS to the international decade for natural hazards reduction. In Proceedings of the Fourth International Conference on Seismic Zonation (Stanford University: Earthquake Engineering Research Institute) 3, pp. 445± 468. de S ilva, F ., P idd, M ., and E glese, R ., 1993, Spatial decision support systems for emergency planning: an operational research /geographical information systems approach to evacuation planning. In Proceedings of the 1993 Simulation Multiconference on the International Emergency Management and Engineering Conference (San Diego: The Society for Computer Simulation), pp. 130± 133. D unn, C . E ., 1992, Optimal routes in GIS and emergency planning applications. Area, 24, 259± 267. E mani, S ., R atick, S . J ., C larke, G . E ., and D ow, K ., 1993, Assessing vulnerability to extreme storm events and sea-level rise using geographical information systems (GIS). In Proceedings of GIS /L IS `93 , (Bethesda, Maryland: American Society for Photogrammetry and Remote Sensing), pp. 201± 209. E mmi, P . C ., and H orton, C . A ., 1995, A Monte Carlo simulation of error propagation in a GIS-based assessment of seismic risk. International Journal of Geographical Information Systems, 9, 447± 46. E stes, J . E ., M c G uire, K . C ., F letcher, G . A ., and F oresman, T . W ., 1987, Coordinating hazardous waste management activities using geographical information systems. International Journal of Geographical Information Systems, 1, 359± 386. E vans, J . R ., and M inieka, E ., 1992, Optimization algorithms for networks and graphs (New York: M. Dekker). FEMA 1984, Application of the I-DY NEV system. Five demonstration case studies (Washington, DC: Federal Emergency Management Agency REP-8). F lowerdew, R ., and G reen, M ., 1992, Developments in areal interpolation methods and GIS. Annals of Regional Science, 26, 67± 78. F otheringham, A . S ., and R ogerson, P . A ., 1993, GIS and spatial analytic problems. International Journal of Geographical Information Systems, 7, 3± 19. G atrell, A . C ., and V incent, P ., 1991, Managing natural and technological hazards. In Handling Geographical Information: Methodology and Potential Applications . Edited by I. Masser and M. Blakemore (London: Longman), pp. 148± 180. G lickman, T . S ., 1986, A methodology for estimating time-of-day variations in the size of a population exposed to risk. Risk Analysis, 6, 317± 324. G oodchild, M . F ., 1987, A spatial analytic perspective on geographical information systems. International Journal of Geographical Information Systems, 1, 327± 334. G oodchild, M . F ., 1992, Geographical data modelling. Computers & Geosciences, 18, 401± 408. G oodchild, M . F ., K linkenberg, B ., and J anelle, D . G ., 1993, A factorial model of aggregate spatio-temporal behaviour: application to the diurnal cycle. Geographical Analysis, 25, 277± 294. H an, A ., 1990, TEVACS: Decision support system for evacuation planning in Taiwan. Journal of T ransportation Engineering-ASCE, 116, 821± 830.
Modelling evacuation vulnerability
783
H obeika, A . G ., and J amei, B ., 1985, MASSVAC: a model for calculating evacuation times
under natural disasters. Proceedings of the Conference on Emergency Planning, Emergency Planning, Simulation Series, Volume 15, Number 1, edited by J. M. Carroll ( La Jolla: Society for Computer Simulation), pp. 23± 28. H obeika, A . G ., R adwan, A . E ., and J amei, B ., 1985, T ransportation Actions to Reduce Evacuation T imes under Hurricane /Flood Conditions: A Case Study of V irginia Beach City . Department of Civil Engineering, Virginia Polytechnic Institute and State University, Blacksburg, Virginia. H obeika, A . G ., K im, S ., and B eckwith, R . E ., 1994, A decision support system for developing evacuation plans around nuclear power stations. Interfaces, 24, 22± 35. H odgson, M . E ., and P alm, R ., 1992, Attitude toward disaster: a GIS design for analysing human response to earthquake hazards. Geo Info Systems, July± August , 41± 51. H onneycut, D ., 1996, Personal communication. Santa Barbara, California, December. J in, L . M ., and C han, S . P ., 1992, A genetic approach for network partitioning. International Journal of Computer Mathematics , 42, 47± 60. J ohnson, D . S ., A ragon, C . R ., M c G eoch, L . A ., and S chevon, C ., 1989, Optimization by simulated annealing: an experimental evaluation; part I, graph partitioning. Operations Research, 37, 865± 892. J ohnson, G . O ., 1992, GIS applications in emergency management, URISA Journal, 4, 66± 72. K ernighan, B . W ., and L in, S ., 1970, An e Y cient heuristic procedure for partitioning graphs. Bell Systems T echnical Journal , 49, 291± 307. L aguna, M ., F eo, T . A ., and E lrod, H . C ., 1994, A greedy randomized adaptive search procedure for the two-partition problem. Operations Research, 42, 677± 687. M c M aster, R . B ., 1988, Modelling community vulnerability to hazardous materials using geographic information systems. In Proceedings of the T hird Symposium on Spatial Data Handling , pp. 143± 156. (Columbus: International Geographical Union). N ewsom, D . E ., M adore, M . A ., and J aske, R . T ., 1992, Evacuation Modelling Near a Chemical Stockpile Site. Argonne National Labs, Document ANL /CP-73412, Argonne, Illinois. NRC 1980, Criteria for preparation, evaluation of radiological emergency response plans and preparedness in support of nuclear power plants . U.S. Nuclear Regulatory Commission, NUREG-0654, Washington, D. C. OFD 1992, T he Oakland tunnel ® re, October 20, 1991: a comprehensive report prepared by the Oakland Fire Department . Oakland, California. P arrot, R ., and S tutz, F . P ., 1991, Urban GIS applications. In Geographical Information Systems, V olume 2: Applications, edited by D. J. Maguire, M. F. Goodchilld, D. W. Rhind, ( London: Longman) 247± 260. P erry, R ., 1985, Comprehensive Emergency Management: Evacuating T hreatened Populations ( London: JAI Press, Inc.). P idd, M ., de S ilva, F . N ., and E glese, R . W ., 1996, A simulation model for emergency evacuation. European Journal of Operations Research, 90, 413± 419. P irkul, H ., and R olland, E ., 1994, New heuristic solution procedures for the uniform graph partitioning problem: extensions and evaluation. Computers and Operations Research, 21, 895± 907. R adke, J ., 1995, Modelling urban /wildland interface ® re hazards within a geographical information system. Geographic Information Sciences, 1, 9± 21. R ejeski, D ., 1993, GIS and risk: a three culture problem. In Environmental Modelling with GIS, edited by M. F. Goodchild, B. O. Parks, and L. T. Steyaert, ( New York: Oxford University Press), pp. 318± 331. S heffi, Y ., M ahmassani, H ., and P owell, W . B ., 1982, A transportation network evacuation model. T ransportation Research A , 16A, 209± 218. S hu- Q uiang, W ., and U nwin, D . J ., 1992, Modelling landslide distribution on loess soils in China: an investigation. International Journal of Geographical Information Systems, 6, 391± 405. S inuany-S tern, Z ., and S tern, E ., 1993, Simulating the evacuation of a small city: the e Vects of tra Y c factors. Socio-Economic Planning Sciences, 27, 97± 108. S orensen, J . H ., C arnes, S . A ., and R ogers, G . O ., 1992, An approach for deriving emergency planning zones for chemical munitions emergencies. Journal of Hazardous Materials, 30, 223± 242.
784
Modelling evacuation vulnerability
S orensen, J . H ., V ogt, B . M ., and M ileti, D . S ., 1987 Evacuation: an assessment of planning and research, Oak Ridge National Laboratory, ORNL-6376, Tennessee. S outhworth, F ., 1991, Regional evacuation modelling: a state-of-the-art-review. Centre for Transportation Analysis, Oak Ridge National Laboratory, ORNL/ TM-11740. S outhworth, F ., and C hin, S . M ., 1987, Network evacuation modelling for ¯ ooding as a result of dam failure. Environment and Planning A , 19, 1543± 1558. S tone, J . R ., 1983, Hurricane Emergency Planning: Estimating Evacuation T imes for Nonmetropolitan Coastal Communities, Sea Grant College Publication, UNC-SG-83± 2,
North Carolina State University, Raleigh.
T ufekci, S ., and K isko, T . M ., 1991, Regional evacuation modelling system ( REMS): A decision support system for emergency area evacuations. Computers & Industrial Engineering , 21, 89± 93. U rbanik, T . E ., and J amison, J . D ., 1992, State of the art in evacuation time estimate studies for nuclear power plants . NUREG / CR-4831, PNL-7776, U.S. Nuclear Regulatory
Commission, Washington, D.C.
U rbanik, T ., D esrosiers, A ., L indell, M . K ., and S chuller, C . R ., 1980, Analysis of techniques for estimating evacuation times for emergency planning zones. O Y ce of Nuclear Reactor Regulation, U.S. Nuclear Regulatory Commission, NUREG /
CR-1745, Washington, DC.
V ogt, B . M ., and S orensen, J . H ., 1992, Evacuation Research: A Reassessment . Oak Ridge National Laboratory, ORNL/ TM-11908, Tennessee. W adge, G . 1988 The potential of GIS modelling of gravity ¯ ows and slope instabilities. International Journal of Geographical Information Systems, 2, 143± 152. WSA (Wilbur Smith and Associates), 1974, Roadway Network and Evacuation Study, Seabrook, New Hampshire, Report prepared for the Public Service Company of New Hampshire,
New Haven, Connecticut, December.
